Work Out Mathematics for Economists

282 Pages·1986·23.262 MB·English

by  A. J. Mabbett (auth.)

MACMILLAN
MASTER
SERIES
Work Out
Mathematics
for Economists
The titles
in this
.
sertes
For examination at 16+
Biology Mathematics
Chemistry Physics
Computer Studies Principles of Accounts
English Language Spanish
French Statistics
German
For examinations at 'A' level
Applied Mathematics Physics
Biology Pure Mathematics
Chemistry Statistics
English Literature
For examinations at college level
Operational Research
Mathematics for Economists
MACMILLAN
MASTER
SERIES
A. J. Mabbett
M
MACMILLAN
© A. J. Mabbett 1986
All rights reserved. No reproduction, copy or transmission
of this publication may be made without written permission.
No paragraph of this publication may be reproduced, copied
or transmitted save with written permission or in accordance
with the provisions of the Copyright Act 1956 (as amended).
Any person who does any unauthorised act in relation to
this publication may be liable to criminal prosecution and
civil claims for damages.
First published 1986
Published by
MACMILLAN EDUCATION LTD
Houndmills, Basingstoke, Hampshire RG21 2XS
and London
Companies and representatives
throughout the world
Typeset by TecSet Ltd,
Wallington, Surrey
British Library Cataloguing in Publication Data
Mabbett, Alan
Work out mathematics for economists.-(Work out
series)
1. Economists, Mathematical
I. Title II. Series
510'.2433 HB135
ISBN 978-0-333-38817-4 ISBN 978-1-349-07863-9 (eBook)
DOI 10.1007/978-1-349-07863-9
To Sandra, Richard and Claire- thank you for all your patience
Preface vii
1 How to Survive the Course 1
1.1 Introduction 1
1.2 The Fear of Mathematics 2
1.3 Study Skills 2
1.4 Some Points to Remember 3
1.5 The Way Forward 3
2 Revision of Basic Algebra 5
2.1 Introduction 5
2.2 The Number System 5
2.3 The Components of an Equation 6
2.4 Equalities and Inequalities 7
2.5 Subscripts,~ (Sigma), 1r (Pi) and Absolute Value 9
2.6 Indices 11
2.7 Logarithms 12
2.8 Rules for Algebraic Manipulation 12
2.9 Examples of Algebraic Manipulation 14
2.10 Sets, Relations and Functions 15
2.11 Some Common Graphs 16
2.12 Equilibrium 22
2.13 Series 25
3 Algebra in Operation 26
3.1 Detailed Solutions to Typical Problems 26
3.2 Outline Solutions to Further Problems 35
3.3 Practice Problems 42
3.4 Answers to Practice Problems 43
4 Differentiation of a One-Variable Function 51
4.1 Review of the Rules of Differentiation 51
4.2 Detailed Solutions to Typical Problems 58
4.3 Outline Solutions to Further Problems 69
4.4 Practice Problems 76
4.5 Answers to Practice Problems 78
5 Maximisation and Minimisation 84
5 .1 Review of the Criteria for Optimisation 84
5.2 Detailed Solutions to Typical Problems 86
5.3 Outline Solutions to Further Problems 103
5.4 Practice Problems 112
5.5 Answers to Practice Problems 114
v
6 Differentiation of Multivariate Functions 120
6.1 Review of Partial and Total Differentiation 120
6.2 Detailed Solutions to Typical Problems 127
6.3 Outline Solutions to Further Problems 135
6.4 Practice Problems 140
6.5 Answers to Practice Problems 141
7 Unconstrained Extrema 145
7.1 Review of the Criteria for the Optimisation 145
of a Function of Two Variables
7.2 Detailed Solutions to Typical Problems 148
7.3 Outline Solutions to Further Problems 161
7.4 Practice Problems 170
7.5 Answers to Practice Problems 171
8 Constrained Extrema 176
8.1 Review of the Lagrange-Multiplier Method 176
for a Function of Two Variables
8.2 Detailed Solutions to Typical Problems 179
8.3 Outline Solutions to Further Problems 192
8.4 Practice Problems 201
8.5 Answers to Practice Problems 202
9 Integral Calculus 208
9.1 Review of the Indefinite and Definite Integral 208
9.2 Detailed Solutions to Typical Problems 215
9.3 Outline Solutions to Further Problems 224
9.4 Practice Problems 229
9.5 Answers to Practice Problems 230
10 Matrix Algebra 235
10.1 Review of the Algebra of Matrices 235
10.2 Detailed Solutions to Typical Problems 245
10.3 Outline Solutions to Further Problems 254
10.4 Practice Problems 262
10.5 Answers to Practice Problems 263
Further Reading 269
Index 271
I have written this book specifically for those students studying an introductory
course in mathematics; a course which is related to economics, business studies or
accountancy. The need for such a workbook as this has become increasingly
apparent in recent years with a substantial number of students having only a very
elementary grounding in mathematics. Many such students lack that initial con
fidence of tackling such 'applied' questions as are commonly found in introductory
economics. The aim of this book is to get such students started on the road to
problem solving.
I should like to thank those students who have inspired me to put all my verbal
explanations onto paper. Without their constant encouragement, I would have
never started such a project as this. My thanks also go to them for trying out
many of the problems. I am not conscious of having made any direct use of
other people's examples; however, over the years it is only too easy to collect
problems from different sources only to forget their origin; and I apologise if this
has happened inadvertently. I should like to thank my colleagues for their con
stant advice and encouragement. Lastly, my gratitude to my family for their
patience while Dad disappeared to write 'that dreaded book'.
Bromsgrove, 1986 A.J.M.
vii
1 How to Survive
the Course
1.1 Introduction
Hello. You are probably wondering what sort of book this is. Well, I am wonder
ing what sort of reader you are going to turn out to be. Let me guess first and
then I will answer your question.
You are most likely to be a first-year student in a Polytechnic or University
studying for a degree in economics, business studies or accountancy. You have
just faced the first few lectures on a course in Quantitative Methods and are
wondering what you have let yourself in for. You have probably received some
problem sheets, had a go at a few questions and got stuck. You have perhaps
turned to some of your fellow students and found that they are in a similar position.
You don't feel that you can go to the tutor and show your ignorance, so you are
now in a state of near panic. Perhaps I am exaggerating, but I have seen the signs
and symptoms a great number of times before. Even students who have a good
Advanced Level pass in mathematics sometimes become unstuck when faced with
an 'applied' question. On the other hand, you could be a student who is either having
to study at home or is only attending on a part-time basis. In which case, you are
looking for a helping hand. If it is near examination time, then you may be a
student who is looking for some extra assistance with revision. Perhaps you are a
mature student who hasn't faced any mathematics for years, but now feels the
need for a refresher course. Furthermore, you prefer to see the applications of
mathematics rather than get bogged down in the theory. You may even be a fellow
lecturer who is looking for a text to recommend to your students, something that
will enhance all your hard work in the lecture room. Equally, you may simply be
looking for more problems to give to your students, especially something that will
avoid the necessity of having to work out all those solutions. Whichever type of
reader you turn out to be, then I hope you will not be disappointed in what you
find in the following pages.
Now for your question. First of all, this is not a textbook. So if you are looking
for a book that will explain all the intricacies of calculus then you haven't found
it here. What you have here is more like your very own tutor. This book is to be
viewed as a friend, something to have by your side helping you through a difficult
period in your education. Having faced a number of cohorts of students on an
economics degree and experienced their frustrations and worry at not being able
to cope with the mathematics, I felt that a text such as this would be of great
benefit. I find the need even more apparent when I try to spread myself around a
large seminar group in the attempt of providing a personal clinic (because nearly
everyone has a different problem!). I usually end up by telling the students to
make sure that they have read the recommended text for the course. I then get
the reply 'Well, the text is great for the theory but it doesn't tell me how to get
started on the problems', or 'The examples in the text are fine, but a number of
steps are missed out or the answer is only one line'. All my students have been
the same; they all prefer the more personal approach. Hence this book.
I have covered all the main topics to be found on the mathematics side of a
first-year Quantitative Course - algebra, differentiation of univariate and multi
variate functions, unconstrained and constrained optimisation, integration and
Matrix Algebra. With the exception of Chapters 2 and 3, all the chapters are
divided into five sections. The first section reviews the main elements of the
subject matter; all those concepts which would have been covered in the lecture
programme and described fully in the recommended text for the course. This
section is also ideal for revision. The next section contains four or five typical
problems with detailed answers. The answers may be too detailed for some readers
because I have tried to spell out all the steps that should be taken. The next
section again has four or five problems, but this time the answers are slightly less
detailed. The penultimate section contains a number of practice problems cover
ing the sort of topics you could get examined on. The answers to these problems
are given in the last section of the chapter; I have avoided the one-line answer, so
even here you will get enough information to be able to follow the workings
through. Chapter 2 is meant to be a revision section for those of you who have
some trouble with basic algebra. For others it could easily be used as an aide
memoire. Chapter 3 contains examples on the application of the algebraic ideas
covered in Chapter 2. Many of the problems in this book have been used either as
examination questions or in a seminar context, so they should be reasonably well
tried. However, some mistakes may still be present, so please accept my apologies
if you find any. I just hope that they are an extremely rare event.
1. 2 The Fear of Mathematics
Students embarking upon a course in economics or related subjects will invariably
have very different levels of mathematical skills. Many will have either a poor
mathematics background or will have forgotten much of what they were taught.
It is therefore not surprising that when faced with the realisation that some
mathematics is necessary, indeed essential, for the study of economics, anxiety
bordering on panic soon sets in. Questions such as 'How much mathematics do I
have to know?' and 'Will I be able to cope?' start going through their minds. The
answers are reassuring. Most undergraduate courses in economics only expect a
basic level of mathematical skill; this will be enough to gain a reasonable under
standing of the economics. My experience and the experience of colleagues has
shown that the majority of students with only a limited mathematical background
are capable of achieving an acceptable level of proficiency in mathematics by the
end of the course. Indeed, I have seen some very marked improvements in suppos
edly weak students. So, DO NOT WORRY. There have been a large number of
students who have successfully trodden this same path before.
1. 3 Study Skills
How to learn is something which is very difficult to teach. When faced with a new
group of students, the first thing I try to stress is ATTITUDE. Without a positive
mental attitude to the course and in particular towards mathematics, then it is
2